By Proposition REF , we have
\(\lbrace L_{mi}\ast A,B\rbrace =\lbrace L_{mi}\boxtimes A,\mathrm {Res}^{\alpha ^{\prime }}_{mi,\alpha }B\rbrace =\lbrace L_{mi}\boxtimes A,L_{mi}\boxtimes {_{mi}\mathcal {R}}(B)\rbrace .\) 
By Section 8.1.10(f) in [1]}, we have
\(\lbrace L_{mi}\boxtimes A,L_{mi}\boxtimes {_{mi}\mathcal {R}}(B)\rbrace =\lbrace L_{mi},L_{mi}\rbrace \lbrace A,{_{mi}\mathcal {R}}(B)\rbrace .\) 
Lemma 1.4.4 in [1]} implies that
\(\lbrace L_{mi},L_{mi}\rbrace =\prod _{k=1}^{m}\frac{1}{1-v^{2k}}.\) 
Hence, we get the first desired result.
The second one is similar.
